Growing up in the Uruguayan countryside with little exposure to art, Alpuy first began drawing at the age of twenty. Within a year, the young artist met Joaquín Torres-García. Inspired by the master's theories of universal constructivism, Alpuy joined the Taller Torres- García, and is today recognized as one of the Taller's most important members.

In 1944, Alpuy contributed two murals as part of the Taller's project to decorate the St. Bois hospital in Uruguay; he would continue to create murals throughout his career. Encouraged by Torres-García, Alpuy and other Taller members travelled to the Andean region of South America in 1945; this experience, along with other periods of travel during the 1950s in South America, Europe, and the Middle East profoundly affected the themes and structural composition of his art.

In 1961, Alpuy emigrated to New York, where he remained for the duration of his life. Alpuy's art has been featured in numerous exhibitions about the Taller Torres-García, as well as in several international one-person exhibitions. His works are included in major international collections including the Museum of Fine Arts, Houston, Texas; The Morgan Library and Museum, New York; The Nelson A. Rockefeller Collection, New York; and Museo Nacional de Artes Visuales, Montevideo, Uruguay.